General Literature on Numerical Methods

Conte, S. D., de Boor, c.: Elementary Numerical Analysis, an Algorithmic Approach. New York: McGraw-Hill 1972.

Dahlquist, G., Bjorck, A.: Numerical Methods. Englewood Cliffs, N.J.: Prentice-Hall 1974.

Forsythe, G. E., Malcolm, M. A., Moler, C. B.: Computer Methodsfor Mathematical Computations. Englewood Cliffs, N.J.: Prentice-Hall 1977.

Froberg, C. E.: Introduction to Numerical Analysis. 2nd edition. Reading, Mass.: Addison-Wesley 1969.

Gregory, R. T., Young, D. M.: A Survey of Numerical Mathematics. Vols. 1, 2. Reading, Mass.: Addison-Wesley 1972, 1973.

Henrici, P.: Elements of Numerical Analysis. New York: McGraw-Hill 1964. Hildebrand, F. B.: Introduction to Numerical Analysis. 2nd edition. New York:

McGraw-Hill 1974. Householder, A. S.: Principles of Numerical Analysis. New York: McGraw-Hill 1953. Isaacson, E., Keller, H. B.: Analysis of Numerical Methods. New York: Wiley 1966. Milne, E. W.: Numerical Calculus. 2nd edition. Princeton, N.J.: Princeton University

Press 1950. Ralston, A., Rabinowitz, P.: A First Course in Numerical Analysis. New York:

McGraw-Hill 1978. Rutishauser, H.: Vorlesungen uber Numerische Mathematik. Vols. 1,2. Edited by M.

Gutknecht in collaboration with P. Henrici et al. Basel: Birkhauser 1976. Scarborough, J. B.: Numerical Mathematical Analysis. 6th edition. Baltimore: Johns

Hopkins Press 1966. Stiefel, E.: An Introduction to Numerical Mathematics. New York, London:

Academic Press 1963. Todd, J.: A Survey of Numerical Analysis. New York: McGraw-Hill 1962. ---: Basic Numerical Mathematics. Vol. I. Numerical Analysis. Basel: Birkhauser

1978 (also New York: Academic Press 1978). ---: Basic Numerical Mathematics. Vol. 2. Numerical Algebra. Basel: Birkhauser

1977 (also New York: Academic Press 1977). Werner, H.: Praktische Mathematik I. Methoden der Linearen Algebra. Berlin,

Heidelberg, New York: Springer-Verlag 1970. ---, Schaback R.: Praktische Mathematik 1I. Methoden der Analysis. Berlin,

Heidelberg, New York: Springer-Verlag 1972.

597

Index

In general, page numbers in italics refer to definitions.

Abramowitz 148 absolute error 12, 279 absolutely continuous 88, 94 absolutely stable 464 absolute norms 388 A-conjugate 305, 573 Adams - Bashforth methods 430ff Adams-Moulton methods 43Off,450 ADI-methods 563, 564ff, 577, 585, 587

convergence of 565ff optimal parameters of 569ff

Ahlberg 94 Aitken's ~2-method 154, 296ff Aitken's interpolation algorithm 43,45, 295 algebraic multiplicity 316, 320ff algorithm 9

graph of 19 numerically more trustworthy 17 numerically stable 18 well behaved 18

alternating-direction implicit,(see ADI) amplification of errors 14 approximation error 1,2,57, 119, 120, 121 approximation method 90 approximation through interpolation 51 arithmetic

complex 285 fixed-point 3 floating-point 5, II interval 27, 28 statistical error 32

arithmetic operations 7 artificial variables 235 Ashenhurst 5 associative law of arithmetic 8 asymptotic expansion 1I8,132, 136,140,420

of central difference quotient 137 of global discretization error 419ff, 450ff of trapezoidal sum 117

attenuation factors 91 Axelsson 575

Babtist 308 BabuSka 489 back-substitution 160

round-off errors in I 88ff backward analysis of errors 19, 27, 489 backward deflation 278 badly conditioned 289 Bairstow's method 285ff Banachiewicz' method 167 band matrix 168, 373, 514ff band width 373 Barker 537 Bartels 213 Barth 352 basic (feasible) solution 224 basis of a linear program 224 Bauer 18, 19, 132, 136, 270, 385, 388 Bendixson, theorem of 395 benign 19

599

600

Bernoulli numbers 12 7 Bernoulli polynomials 129 Bessel functions 107 BFGS-method 302, 306, 308 bidiagonal matrix 379, 380, 382 bilinear form 508[f, 520 bisection method 149,283, 352, 469 bit 3,4 bit-cycling permutation 112 bit-reversal permutation 81 Bjorck 597 block Gauss-Seidel method 561 block iterative methods 560 block Jacobi methods 562 block relaxation methods 562 block single-step method 561 block total-step method 561 Bloomfield 78 Bohmer 93 boundary conditions

nonlinear, separated 466 boundary-value problems 93,405,466,476,

488 linear 475

Ciarlet 57, 515, 525 Clark 462 Coddington 409 Collatz 414,498 column sum criterion 542 complete pivot selection 161, 187, 188 complex arithmetic 285 complex conjugate roots 285 composite of integration rule 121 computers (digital, analog) 2, 3 cond 179 condition

of matrix 179 of eigenvalue problem 334ff, 358, 391 of linear equations 175ff of linear least squares 202ff

condition numbers 13, 306 conjugate-gradient method 537,572,573,

576,585 for least squares 576

consistently ordered 548, 549ff consistent matrix norm 177 constrained minimization 245 Conte 597

model problem for 555, 566ff, 57Off, 583ff continuation methods 488ff singular 489, 490 continued fraction 65 with free boundary 468, 494 contraction property 251

Bowdler 376 contractive mapping 251, 300 Brent 295 Brigham 78 Broyden 302 Broyden, Fletcher, Goldfarb, Shanno (see

BFGS) Broyden's rank-one method 268,302, 487 B-splines 114 Bulirsch 72, 86, 102, 134, 135, 136, 138,

139,153,460,483,492,517,518 Bunch 159 Buneman 577

algorithm of 537, 576ff, 583 suggestion by 581

Butcher 415 Buzbee 577,583,584

cancellati on 8, 14 Cauchy's convergence criterion 248 Cayley-Hamilton Lemma 323 central difference quotient 13 7 characteristic polynomial 149, 27Off, 283,

290, 316ff, 345ff, 352ff, 395, 440 Chebyshev polynomials 148, 157 Chebyshev system 145 Choleski decomposition 172ff, 304, 368, 377,

382ff, 506, 512, 557, 561, 565

convergence alternating 250 cubic 374 essential 372 global 249 linear 250, 292 local 249 quadratic 250, 254, 274 superlinear 307

convex 253 Cooley 78 Cooley - Tukey method 78, 111, 112 corrector methods 431, 449, 455, 456 Courant, Weyl, theorem of 394 covariance matrix 20 I Cramer's rule 353 Crane 462 Crout's method I 66ff cubic convergence 374 cubic splines 38, 93, 513 curvature 97

Dahlquist 449, 597 damping of errors 14, 29, 47 Daniel 214, 217 Dantzig 222, 225, 235

Index

data fitting I 97ff Davenport 462 Davidon 302, 307 Davidon, Fletcher, Powell (see DFP) Davis 38, 118, 123 de Boor 93,97, 102,597 decimal number representation 3, 4 decomposition

QR 190ff triangular 164ff

decomposition of maps 10 definite integral 117 deflation 278, 296

backward 278 forward 278, 279

degenerate linear program 224 degree type of rational interpolation

problem 58 Dekker 291 ~2-method of Aitken 154, 296ff Dennis 269 derogatory matrix 322 descent direction 301 determinant 167 Deuflhard 483,489,492, 518 DFP-method 302, 308 diagonalizable matrix 321ff, 329ff diagonally dominant 542

irreducibly 543 diagonal matrix 184, 321, 342, 402 diagram of degree types 62 Diekhoff 462 difference equations 437ff

linear 438 stability condition for 439

difference methods 502, 536 for ordinary differential equations 502, 504,

507,516,518 for partial differential equations 554ff

difference quotient central 137, 141 one-sided 137

differential equations ordinary 93, 128, 404 ff, 502ff, 516ff partial 93, 554ff

differential error analysis 12 differential operator 508

positive definite 509 symmetric 508, 520

digits 1 leading 8 significant 4

direct methods for linear equations 159ff, 190ff

Dirichlet boundary-value problem 519, 554

601

discretization (discretizing) 1, 117ff, 136ff, 576

discretization error Iff global '415ff, 419, 450 local 413, 433

distributive law of arithmetic 8 divided differences 43, 44ff, 48, 50, 54, 63,

295 divided-difference scheme 45, 49 dividing off 278, 281 Dixon 306 Dongarra 159 Dorr 584 double-step method 274ff, 279

Eberlein 345 eigenproblem, generalized 307 eigensolution 314 eigenvalue problem 314

algebraic 314 for differential equations 467 generalized 382

eigenvalues 149, 150,270,283,314,315, 316ff, 351, 383

algebraic multiplicity of 316, 320ff exclusion theorems for 383 extremal properties for 394 geometric multiplicity of 317, 320ff ill-conditioned 358, 391 multiplicity of 316, 319ff numerically acceptable 357ff sensitivity of 388 well-conditioned 356

eigenvector 149, 150,315, 316ff left 318, 372, 389ff numerically acceptable 357, 359ff

elementary divisors 321, 324, 357ff, 391 inequalities 222 maps /0, 17 operations 9, IS, 32

elimination Gaussian 159ff Gauss-Jordan 169ff

elimination matrix 214, 348 end corrections 123 Enright 462, 465 equilibrated matrix 161, 184ff equivalent rational expressions 59 Erdelyi 132 error

absolute 12 approximation Iff, 57, 119ff discretization Iff, 413, 415ff, 419

602

error [cont.J inherent 18 input 1 integration 123 interpolation 57 relative 12 round-off Iff truncation Iff

error amplification 14 error analysis

backward 19, 27 differential 12

error bounds for Hermite interpolation 57 for linear equations 175ff for linear least squares 20 Iff for spline interpolation 105ff

error damping 14, 29,47 error distribution 30, 32 error propagation 11, 18, 20, 31 essential convergence 372 Euclidean algorithm 282 Euclidean (vector) norm 176, 262, 306, 335 Euler-MacLaurin summation formula 118,

122, 127ff, 132 Euler's method 411ff, 418, 420, 451, 462ff

implicit 463ff modified 414ff region of absolute stability for 464

evaluation of characteristic polynomials 272 of polynomials 44 of trigonometric expressions 84ff

exact line search 260,304ff exclusion theorems 383 exponent 4

over(under) flow 6, 7 exponential interpolation 38 exponential spline 115 extended integration rules 121 extrapolation 51 extrapolation methods 38,71, 118, 128

for differential equations 420 for differentiation 137 for initial value problems 458, 460, 463,

471,517 for integration 131ff, 151

fast Fourier transforms 38, 78ff feasible point 222 Fehlberg 415,426,427 Fellen 462 Fibonacci sequence 527 Fike 385, 388

Index

finite element method 519, 523, 526, 536 Fix 526 fixed point 248 fixed-point representation 3, 4 Fletcher 302 floating-point arithmetic 4ff, 15ff, 420ff floating-point operations 7ff floating-point representation 4ff

normalized 4 ff unnormalized 5

Forsythe 540, 597 forward deflation 278ff Fourier

analysis 37 coefficients 88 series 88 synthesis 84 transform 37, 78

Fox 462 fractional iteration 356 Francis 315, 370, 376

observation of 380ff Frazer, Duncan, Collar, method of 346 free variable 223 Frobenius matrices 161,324, 326ff, 395,440 Frobenius normal form 324,326, 345 Froberg 553, 597 fully nondegenerate rational interpolation

problem 62

Garbow 315 Gass 222, 235 Gauss' arithmetic-geometric mean

algorithm 569 Gaussian

elimination 102, 159ff, 183ff, 349, 361, 486

integration (quadrature) 118,121,142,151, 153

Gauss-Jordan method 169ff Gauss-Newton method 210ff Gauss-Seidel method 539, 544ff, 547, 553,

557ff, 560, 585, 587 Gautschi 91, 148 Gear 406,458,465 generalized

eigenproblem 307 inverse 210ff Lagrange polynomials 53 spline functions 97

Gentleman 78 geometrically increasing step length 139 geometric multiplicity 317, 320ff George 526, 534, 584

Index

Gershgorin's theorem 385ff, 504 Gill 214 Ginsburg 585 Givens 403

method of 341 ff reflection 216, 379ff rotation 216, 342, 373

global discretization error 415ff, 419, 450 globally convergent 249 Goertzel 84 Goldfarb 302 Goldstein 7 Golub 148, 149, 150, 214, 270, 377, 382 Golub and Reinsch, method of 315,378,381 Gordon 406,458 gradient 257 Gragg 214,217,419,453,454 Gragg's function 458 Gram - Schmidt orthogonalization 143, 195ff

with reorthogonalization 197 graph of algorithm 19 graph of matrix 542

connected 542, 558 (weakly) 2-cyclic 551, 558

great circle distance 35 Green's formula 520 Gregory 597 Greville 93, 97 Grigorieff 406,415, 462, 465 Grobner 117 Grossman 198 Guest 198

Haar condition 145 Hackbusch 584, 588 Hadley 222, 235 Hall 106 Hamilton (see Cayley - Hamilton) harmless round-off errors 18 Handbook of Mathematical Functions 148 Hanson 199 Hausdorff 393 Helmholtz equation 584 Henrici 123,270,281,406,407,597 Hermite

function space 56, 515 interpolation 49,52ff, 120, 126 polynomials 148, 154

Hermitian matrix 153,315, 328ff, 394 Herriot 102 Hessenberg matrix 187,218,315,335,339,

349ff. 353. 361 Hessian matri x 269 Hestenes 493. 572

Heun 414 method of 414ff

Hildebrand 597 Himmelblau 245 Hindmarsh 465 Hirsch 383, 395 Hockney 577 Hofreiter 117 Holladay's identity 94 homogeneity axiom 176 homotopy methods 488ff,491 Homer scheme 44,84,271,287 Householder 191,274,388,392,597

matrix 193 method of 336, 34lff, 345, 351, 378 orghogonalization 191 ff reduction 340, 350

Householder transformation for arbitrary matrix 339 for symmetric matrix 338

Hull 462, 465 Hyman 353

method of 353

ill-conditioned 13 implicit methods for differential

equations 431, 463 improper integrals 154 inaccessible points 61 inclusion theorems 383. 391 indefinite integral 117 inequalities

elementary 222 linear 222

inexact line-search 260 initial-value problem 404,407,410 initial (starting) values

603

rules for the selection of 272, 278, 279 integral

definite 117 improper 154 indefinite 117

integration by parts 88, 89, 128, 130 error 123

integration rules composite (extended) 121

general form of 123 of Gauss 118, 121,142, 151 of Milne 120 of Newton-Cotes 118, 12Off, 126, 133,

136, 151. 429, 434 of Simpson 117, 120. 122. 124. 126, 133 of Weddle 120

604

integration rules [cont.J % 120 trapezoidal 120, 124

interpolation exponential 38 Hermite 49, 52ff, 120, 126 inverse 295 linear 37 polynomial 37ff rational 38, 58ff spline 38 trigonometric 72ff

interpolation error 49, 57 interpolation formula

of Lagrange 39, 52, 118, 138 of Newton 62ff, 294, 295

interpolation on product spaces 108 interval arithmetic 27, 28 inverse differences 63ff inverse iteration of Wielandt 351, 354,356,

360, 361 inverse interpolation 295 inverse of matrix 159, 169ff irreducible matrix 542ff, 547, 551 irreducibly diagonally dominant 543 Isaacson 597 iteration function 245, 270 iterative methods 536, 554

construction of 245ff, 537 convergence of 248ff, 300ff, 540ff for linear equations 536 for minimization 245, 256ff, 300ff for roots of polynomials 27O£f for zeros of functions 244ff, 290ff

iterative refinement 537, 539

Jacobian elliptic function 472, 482ff Jacobian matrix 13, 15ff, 247, 253 Jacobi matrix 336 Jacobi method 34lff, 377, 539, 541, 543, 545,

547, 553, 557, 559ff, 585, 587 Jenkins 270 Jordan block 321 Jordan normal form 319,320ff, 384

Kahan, Theorem of 547 Kantorovich (see Newton-Kantorovich) Kaps 464,465 Kaufman 214, 217 Keller 406, 476, 483, 530, 597 knots of splines 93 Krogh 458, 534 Kronrod 151

Krylov sequence 325, 346 Kulish 30 Kutta 414

Ll, Loo norms 222

Index

Lagrange's interpolation formula 39, 52, 118, 138,429

generalized 120 Lagrange polynomials 39, 107

generalized 53 Laguerre polynomials 148, 154 Laplace operator 526 Liiuchly 206 Lawson 199 leading digits 8 least-squares problem 199ff, 202ff, 576

nonlinear 209ff left eigenvector 318, 372, 389ff Legendre polynomials 147 Levinson 409, 533 Lindberg 465, 533 line search 301, 303ff, 308ff

exact 260, 304 inexact 260

linear approximation method 90 linear convergence 249, 292 linear equations

direct methods for 159f£, I90ff elimination methods for 159ff, 190ff iterative methods for 536 positive definite systems of 172ff

linear multistep methods 445ff, 450 linear programming 222ff linearization 247 Lipschitz condition 88, 307, 406ff, 418 Lipschitz continuous 307, 442 local discretization error 413, 433 locally convergent 249 local minimum 249, 304 Lory 462, 533 LP(I,p) 222 LR-method 315, 351,361, 364, 367ff, 370,

373, 377 double step of 370 with shifts 369

lub 178 Luenberger 302

machine numbers 4ff, 8 machine precision 6, 8, 86, 436 Maehly 62, 278 Maehly's modification of Newton's

method 279, 352

Index

Malcolm 597 mantissa 4, 7 Marden 274 Martin 339, 340, 349, 352, 376, 383 matrix

band 168,373,514,515 bidiagonal 379ff, 382 block tridiagonal 550, 562 consistently ordered 548, 549ff derogatory 322 diagonal 184, 321, 342, 402 diagonalizable 321ff, 329 diagonally dominant 542, 543 Frobenius 161, 324, 326ff, 395, 440 Givens 216 Hermitian 173,315, 328ff, 394 Hessenberg 187,218,315,335,339, 349ff,

353,361

605

modification of matrix decompositions 213ff modified Newton method 256ff Moler 159, 383, 403, 540, 597 moments of cubic splines 97 Moore 30 Muller's method 294ff multiple root 294 multistep methods 429, 463

consistency of 434, 446ff convergence of 435, 441, 446ff linear 445ff of order p 434, 448 step control for 455 weakly unstable 454

Murray 214,242,270,312 Murty 235, 242

Householder 191,336,340,347,371, 378ff Na 489,534 irreducible 351,353, 542ff, 547, 551 National Bureau of Standards 148 Jacobi 336 nonderogatory 322ff, 346 normal 315,328,330, 393 normalizable 321, 388 orthogonal 179 permutation 112, 161, 347, 549 positive (semi) definite 172ff, 305, 330,

376, 565, 576 sparse 537, 557 symmetric 111,168,173,283 triangular 160ff, 361ff tridiagonal 187,290,315,336,361,373 unitary 179, 328ff, 371 upper Hessenberg 187,218,335 with property A 548, 549ff, 558

matrix norm 176ff maximum norm 176 method of false positions (see regula falsi) Metropolis 5 Meyer 106 midpoint rule 415,432,434,450,458

modified 458 Milne 62, 597

corrector method of 432, 450 integration rule of 120, 134

Milne-Thompson 66 minimal polynomial 322ff minimization

constrained, linear 222ff constrained, nonlinear 245 unconstrained, general 245, 256ff, 300ff

minimum-norm property 94 minimum point 198, 209, 245, 300

local 301 model problem 555, 566ff, 570ff, 583ff

natural spline 96 neighboring basis 225 Neumann (see von Neumann) Neville's interpolation algorithm 40ff, 43, 45,

67, 132 Newton-Cotes integration formulas 118,

12Off, 126, 133ff, 136, 151,429,434 Newton direction 301 Newton form of polynomial 44,47,55 Newton-Kantorovich, Theorem of 256 Newton-Raphson method 246 Newton's interpolation formula 43,62, 294ff,

455 error term for 50

Newton's method approximate 474 for minimization 256ff for roots of polynomials 270ff for systems of equations 244, 485, 487 for zeros of functions· 246ff, 253ff general 472,498 Maehly's version of 279, 352 modified 256ff, 474, 480, 485, 487,488,

497 Nickel 270 Nielson 577, 583 Nilson 94 nonbasic variable 224 nondegenerate linear program 224 nonderogatory matrix 322ff, 346 norm 13, 176ff

absolute 388 consistent 177 Euclidean maximum 176

606

norm [cont.J

Sobolov 521, 525 submultiplicative 177 subordinate 178

normal equations 576 normal matrix 315,328,330, 393 normalizable matrix 321, 388 normalized floating-point representation 4 numerically stable 18ff, 85, 349, 581 numerically trustworthy 17, 19 Nystrom's predictor method 431

Oberle 462 objective function 222 Oden 526 Oettli 181 Olver 132 one-step methods 410,412

consistency of 413 convergence of 415,416 linear of order p 413 step control for 423, 425ff, 428

operand set 10 opposite terms 79 optimal block relaxation methods 562 optimal parameters of ADI methods 569ff optimal solution of linear program 223 ordinary differential equations 404ff

boundary-value problem for 405 initial-value problem for 404 of first order 405 of mth order 405 systems of 404

Oren 302, 306 Oren - Luenberger class 302, 307ff Ortega 256, 313 orthogonal

basis 75 matrix 191 polynomials 142, 143ff projector 211, 213

orthogonalization Gram-Schmidt 195ff Householder 190ff to solve least squares 201 ff

Osborne 483 Ostrowski 295, 313

, Reich, theorem of 547 overflow of exponent overrelaxation 546, 547 overshooting 274, 278

Index

partially interpolating polynomials 41 partial pivot selection 161, 163, 170, 187ff,

360, 368 Peaceman and Rachford

ADI method of 563ff formula of 593ff

Peano kernel 124, 126 Peano's error representation 117, 123 permutation matrix 112, 161 perturbation of polynomial coefficients 287 Pesch 462 Peters 278,279,281,290,291,361,376,383 phase

one of simplex method 233ff two of simplex method 222ff

phase polynomial 73ff pivot element 161

partial selection of 161, 163, 170, 187ff complete selection of 161, 187ff selection 161, 183ff

Poisson equation 576, 584, 588 discretized 576

polygon method 411 polynomial

characteristic 149, 27Off, 283, 290, 316ff, 345ff, 352ff, 395

minimal 322ff polynomial interpolation 37

error in 49 polynomials

Bernoulli 129 Chebyshev 148 Hermite 148 Lagrange 39 Laguerre 148 Legendre 147, 156

positive (semi) definite 172ff, 305, 330, 376, 565, 576

Powell 302, 307 Prager 181,489 predictor methods 431ff, 449, 455ff principal vector 321 Proskurowski 584 pseudoinverse of matrix 21Off, 333ff

QR decomposition 19Off, 371 QR method l49, 150~ 315, 351, 353, 367ff,

370ff, 373, 376ff, 380, 382 convergence of 371 ff double step of 376 with shifts 374

quadratic convergence 250, 254, 274

Index

quadratic function 304 quadrature (Gaussian, numerical) 117,142 quasi-Newton

equations 301, 303 methods 301ff

quasilinearization 498 QZ method 383

Rabinowitz 118, 123, 597 Rachford (see Peaceman) Rademacher 30 Ralston 597 random variable 30, 201 rank-one modification 217, 302 rank-two modification 270, 300ff rational

functions 58 interpolation 38, 58 normal form 324, 326

rational expressions 59 equivalence of 59 relatively prime 59

Rayleigh quotient 393 Rayleigh - Ritz -Galerkin method 93, 507 Rayleigh - Ritz method 523 reciprocal differences 66 Reddy 526 reduced costs 227 reduction methods 577 region of absolute stability 464 regula falsi 291ff Reich (see Ostrowski) Reid 537, 575 reinitialization 269 Reinsch 86,91,102,159,199,231,315,339,

376, 377, 382, 540, 575, 585 relative error 12 relatively prime 59 relaxation methods 545,546, 585

convergence of 547 optimal 559ff

relaxation parameters 546 optimal 553, 559,562

Rentrop 462, 464, 465 reorthogonalization 197 restricted variables 223 Reutersberg 584, 588 Rheinboldt 245, 248, 256 right eigenvector 316, 372, 389, 390 Romberg 132, 134 Romberg's integration method 118, 132ff,

138

roots of polynomials complex (conjugate) multiple 294 simple 276

Rose 537 Rosenberg (see Stein)

144, 149, 244, 270 285,294

round-off error distribution 30ff round-off errors

in back-substitution 188ff in Fourier synthesis 84ff in Gaussian elimination 183ff in linear least squares 205ff in one step methods 420ff

row sum criteria 542, 543 row sum (matrix) norm 178 r-step method 432

linear 432 Runge 414

607

Runge-Kutta-Fehlberg method 426, 463ff Runge-Kutta method 414ff, 419ff, 422,

425ff,453 Rutishauser 72,102,132,136,315,343,345,

361, 597 Rutishauser's semilogarithmic notation 4

Sande 78 Sande-Tukey method 78, 112 Saunders 214 Sautter 189 scaling 7, 184, 402 Scarborough 6, 597 Schaback 597 Schoenberg 122 Schroder 584, 588 Schultz 57, 93, 515 Schultz' method 310 Schur 328 Schur norm 178 Schur normal form 328 Schur's theorem 330 Schwarz 241,345 Schwarz inequality 510, 521 search direction 257 Seber 198 secant method 292ff Secrest 142 Sedgvnck 458, 462 semilogarithmic notation 4 sensitivity to input perturbation

of linear equations 179ff of linear least squares 202ff of polynomial roots 287ff

608

Seydel 462 Shampine 406,458,462 Shanks 415 Shanno 302 shooting methods

counter 530 internal subdivision in multiple 487 limiting case of mUltiple 498 multiple 483ff, 486ff, 489, 491ff, 498ff,

515, 517 simple 469ff, 474, 480ff, 486ff, 495, 515,

517 starting trajectory 487ff, 494

sign-changes 281 significant digits (bits) 4 similarity transformation 318, 334 simple Kutta formula 527 simplex method

general step 225 phase one 233ff phase two 222ff

Simpson's rule 117,119, 120,124,126,132, 415

single-step method 539 singular value decomposition 332, 377ff singular values 331 ff, 377ff slack variable 222 Smiili 315,339,340,349,361,376 Sobolov norm 521, 525 SOR meiliods 546

optimal 587 sparse matrix 537, 557 spectral radius of matrix 384, 540 Spedicato 306 spline

cubic 38, 93, 513 exponential 115 interpolation 38, 93ff natural 96

stability condition 439ff, 443ff, 446, 448ff stable, numerically 18ff, 85, 349, 581 stationary point 259, 261 steepest descent meiliod 572 Steffensen 119, 126 Steffensen's convergence acceleration 298ff Stegun 148 Stein, Rosenberg, theorem of 545 step lengili, size 123,406, 411

geometrically increasing 139 sequences for extrapolation meiliods 132,

153 Sterbenz 30 Stetter 406, 415 Stewart 159,214,217,383

Stiefel 132, 136, 345, 572, 597 stiff differential equations 462ff

Index

Stoer 86, 135, 139, 153,308,388,460,483, 492,517,518

stopping criterion for extrapolation meiliod 135, 141

Strang 526 Stroud 142 Sturm sequence 281ff, 352 submultiplicative norm 178 subordinate norm 177 successive overre1axation (see SOR) summation formula of Euler and

MacLaurin 118, 122, 127ff, 132 superlinear convergence 307 support

abscissas 37, 121, 138 ordinates 37 points 37

Swarztrauber 577 symmetric matrix 283 symmetric rank-one method of Broyden 302 system of linear equations 145, 159 Szegii 149

Tang 489 Tewarson 537 Thiele's continued fraction 67 ¥a-rule (integration) 120 Todd 597 Tornheim 311 total step method 539 translation invariant 91 trapezoidal

rule for integration 120ff sum 89, 117,121, 123, 128, 131ff, 134ff,

141, 153 Traub 270, 294 triangle inequality 176 triangular decomposition 561

by direct computation 166ff by Gaussian elimination 159ff by orthogonalization 190ff positive definite case 174

triangular matrix 160ff, 361ff tridiagonalization 336 tridiagonal matrix 148,270,290,315,336,

361, 373 trigonometric expression 72 Troesch 481 Trottenberg 584, 588, 596 trustworthy, numerically 17. 19 Tukey 78

Index

unconstrained minimization 245, 256ff, 300ff underflow of exponent 6, 7 underrelaxation 546 unitary matrix 179,328ff, 371 upper Hessenberg matrix 218, 335

Varga 57,515,533,537,545,547,549,562, 563, 566, 568, 569, 590, 593, 594, 596

, Young, theorems of 552ff variable order method 457 variance

of random variable 201 of round-off distribution 32

variational methods 507, 512ff for ordinary differential equations 507,516,

518 for partial differential equations 519

vector iteration 372, 393, 538 simple 351,354, 356 Wielandt's inverse 351,354,356, 360ff

vector norm 176ff Vitasek 489 von Neumann 7

Wachspress 569 Wagschal 525 Walsh 94 Watts 462

Weddle's rule (integration) 120 weight function 142 weights

in Gaussian quadrature .146, 149 in Newton-Cotes formulas 118

Weinstein, theorem of 392 well behaved 18 well-conditioned 13 Welsch 148, 149, 150 Werner 597 Weyl (see Courant) Widlund 584

609

Wilkinson 19,159,167,187,205,206,231, 278,279,281,289,290,291,315,339, 340,345,349,352,361,365,370,374, 376, 383, 540, 575, 585

Willoughby 465, 535, 537 Wittmeyer 538 Witzgall 62, 388 word length, double, triple 3

Young 537,548,563,568,569,594,597 , Varga, theorems of 552ff

zero of function 244ff,29Off zero suppression 279, 296 Zhimal 524, 525, 526